How can I solve the mentioned problem?Please,help.

Question

If,#cosA+cosB+cosC=0# ;then prove that ,#cos3A+cos3B+cos3C=12cosAcosBcosC#

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Answer 1 · 2017-12-20T14:46:56

Given

#cosA+cosB+cosC=0#

#=>cosA+cosB=-cosC#

#=>(cosA+cosB)^3=-cos^3C#

#=>cos^3A+cos^3B+3cosAcosB(cosA+cosB)=-cos^3C#

#=>cos^3A+cos^3B+3cosAcosB(-cosC)=-cos^3C#

#=>cos^3A+cos^3B+cos^3C=3cosAcosBcosC#

#=>4cos^3A+4cos^3B+4cos^3C=12cosAcosBcosC#
Now
#cos3A+cos3B+cos3B#

#=4cos^3A-3cosA+4cos^3B-3cosB+4cos^3C-3cosC#

#=4cos^3A+4cos^3B+4cos^3C-3(cosA+cosB+cosC)#

#=12cosAcosBcosC+3*0#

#=12cosAcosBcosC#